A random sample of 11 hikers produced the following data, where x is the age of the hiker,and y is the maximum number of miles hiked per week. The data are presented below in the table of values: x & y 6 & 19 7 & 26 10 & 35 13 & 34 14 & 36 17 & 45 18 & 44 20 & 54 23 & 48 24 & 67 25 & 53

1. Calculate the means of ( x ) and ( y )<br /> Compute the average of the ( x ) values and the ( y ) values.<br />\[<br />\bar{x} = \frac{1}{11} \sum_{i=1}^{11} x_i = \frac{6 + 7 + 10 + 13 + 14 + 17 + 18 + 20 + 23 + 24 + 25}{11} = \frac{177}{11} = 16.09<br />\]<br />\[<br />\bar{y} = \frac{1}{11} \sum_{i=1}^{11} y_i = \frac{19 + 26 + 35 + 34 + 36 + 45 + 44 + 54 + 48 + 67 + 53}{11} = \frac{461}{11} = 41.91<br />\]<br /><br />2. Calculate the covariance of ( x ) and ( y )<br /> Compute the sum of the products of the deviations of ( x ) and ( y ) from their means.<br />\[<br />\text{Cov}(x, y) = \frac{1}{n-1} \sum_{i=1}^{n} (x_i - \bar{x})(y_i - \bar{y})<br />\]<br />\[<br />\text{Cov}(x, y) = \frac{1}{10} \left[ (6 - 16.09)(19 - 41.91) + (7 - 16.09)(26 - 41.91) + \ldots + (25 - 16.09)(53 - 41.91) \right]<br />\]<br />\[<br />\text{Cov}(x, y) = \frac{1}{10} \left[ -11.09 \times -22.91 + -9.09 \times -15.91 + \ldots + 8.91 \times 11.09 \right] = 125.64<br />\]<br /><br />3. Calculate the variance of ( x )<br /> Compute the sum of the squared deviations of ( x ) from its mean.<br />\[<br />\text{Var}(x) = \frac{1}{n-1} \sum_{i=1}^{n} (x_i - \bar{x})^2<br />\]<br />\[<br />\text{Var}(x) = \frac{1}{10} \left[ (6 - 16.09)^2 + (7 - 16.09)^2 + \ldots + (25 - 16.09)^2 \right]<br />\]<br />\[<br />\text{Var}(x) = \frac{1}{10} \left[ 123.9881 + 82.7281 + \ldots + 79.4081 \right] = 42.09<br />\]<br /><br />4. Calculate the slope ( b ) of the regression line<br /> Use the covariance and variance to find the slope.<br />\[<br />b = \frac{\text{Cov}(x, y)}{\text{Var}(x)} = \frac{125.64}{42.09} = 2.99<br />\]<br /><br />5. Calculate the intercept ( a ) of the regression line<br /> Use the means and the slope to find the intercept.<br />\[<br />a = \bar{y} - b \bar{x} = 41.91 - 2.99 \times 16.09 = -6.16<br />\]<br /><br />6. Form the regression equation<br /> Combine the slope and intercept to form the equation.<br />\[<br />y = a + bx = -6.16 + 2.99x<br />\]